It is easy to write a computer simulation of specific cases, and watch the bell shape emerge out of it. All that tells you is that the mathematicians are telling the truth, but not WHY it happens.
To explain why is difficult in layman's terms except for a few cases (the binomial is the token example for classical CLT).
For the majority of the people I had in the intro statistics courses I TA'ed for, this was enough explanation. In undergrad, we didn't prove Lindeberg-Levy CLT until my third quarter of math stats, and the versions with weaker assumptions we didn't prove until graduate math stats.
When you're struggling to understand the fundamentals of hypothesis testing and simple combinatorics, going through the rigorous process of proving the CLT is daunting, mind-boggling, and largely unnecessary :).
It is slightly easier if you start off assuming Stirling's formula. However there is no need to assume that - you can derive it at the same time. Elsewhere in this thread I gave an outline of how to derive this simple case.
